I echo other responders in saying that concepts related to the nature versus nurture debate are skillfully embedded in this short story and that Baldwin is careful to offer no easy answers in that regard. However, here are a few points you may want to look to for support or refutation no matter which side you take:

URBAN LANDSCAPE (Evidence for Nurture): The way Baldwin paints the harsh urban landscape seems to point toward nurture. As portrayed in the story, Harlem is a barren place of grey concrete, utilitarian high-rise housing projects, and signs of economic hardship. As portrayed by Baldwin, it is not a place suited to fostering delicate young souls. In fact, the narrator seems to feel just as imprisoned by the landscape as Sonny was by his literal jail cell.

CHILDHOOD INNOCENCE/LOSS OF INNOCENCE (Evidence for Nurture): At several points in the story, the narrator describes seeing a glimpse of the former innocent child in the visage of someone who has become worn out and damaged as an adult. For instance, when looking at his brother's face, the narrator observes: "Yet, when he smiled, when we shook hands, the baby brother I’d never known looked out from the depths of his private life, like an animal waiting to be coaxed into the light." At another point in the story, the narrator encounters a young man about Sonny's age who grew up on their block and also succumbed to the same temptations. The narrator has feelings of disgust toward the young man, but when the man grins, the narrator thinks, "It made him repulsive, and it also brought to mind what he'd looked like as a child." These reminders of one's former childhood innocence still lurking behind the worn adult face seem to suggest that social factors, not inherent traits, are the main factors contributing to the drug crisis in the community.

BROTHERS AS FOILS (Evidence for Nature): Due to their parents' deaths when Sonny was still a minor and the narrator's decision to take Sonny in after his prison sentence, he is literally positioned as his "brother's keeper," a reference to Cain and Able from the Bible. Just as in the Biblical story, these two brothers are very much opposites: one is a soulful musician and the other is a practical math teacher. It seems there is some essential difference in the two men's characters that contributed to their different choices and lifestyles.

No matter what side you take in the debate, remember the hopeful note that the story ends on with the salvation that is suggested by the ending scene at the jazz club. The narrator watches Sonny in his natural element and observes him struggling with but mastering his instrument as he plays. It seems that in this scene, the narrator finally understands his brother. Again, Baldwin leaves it beautifully abstract: We do not know whether the struggles Sonny is metaphorically wrestling with through his music are related to his selfhood, his harsh environment, or some combination. However, it is clear that Sonny is an essentially good person, and the story ends on a hopeful note for Sonny's future. While Sonny's status as a musician signifies something about his essential self, it is his instrument that offers him a route to salvation.

It is not uncommon for musicians to use drugs, for while they are under the influence of drugs, they are often extremely creative.  In addition, the lifestyle of a musician makes the use of drugs tempting as they are often up long hours and/or odd hours and taking something helps them perform longer--at least they think so. 

Musicians and other artistic types "march to a different drummer" from the rest of humanity.  Extremely sensitive, the world often becomes more than they can handle and they seek outlets for their storms from within.  Sonny says that he has done heroin because he felt in control when he did.  At any rate, the nature of a musician is no commonplace one.

I think #2 makes a number of extremely valid points. The whole nature/nurture issue is one that has troubled academics for centuries, and I think the author of this short story is very clever to ensure that there is evidence for both. In a sense, we are unable to tell if we are more a product of our genes than our background - both will clearly play an important role in establishing who we are, but for some, arguably, nature will take the bigger role and for others nurture.

I am offering only an opinion because there is evidence in the text to support both points of view.  Implicit in the story is that Sonny has had a problem with addiction and a gift of music since he was young, and that the narrator has always trod a "straight" path.  If Sonny was born with a predisposition to drugs and alcohol, then certainly, that predisposition is "nature."  But we do not know if there is a predisposition in this family, which would be a strong argument for "nature," too.  On the other hand, it is clear that both brothers have grown up in an environment in which drugs and alcohol are easily available and an environment of poverty and, for many, despair.  That argues for "nurture" as the cause of Sonny's problem.

Now, having pointed out bases for both points of view, it is my opinion that given our current understanding of human genetic and behavior, this is a false dichotomy, meaning there is no reason to say anyone is the result of just one and not the other.  People are born with strengths and weaknesses, for sure, but whether these strengths and weaknesses emerge, or the degree to which they emerge, is a function of environment.  We are creatures of both, so the idea of choosing seems to me to be a mistake.  This is a complex area of debate, and you will certainly find people who disagree with my position.